SIAM J. Disc. MATH.
Vol. 4, No. 2, pp. 182-195, May 1991
(C) 1991 Society for Industrial and Applied Mathematics
0O4
TOWARDS A LARGE SET OF STEINER QUADRUPLE SYSTEMS*
TUVI
ETZION-
AND
ALAN
HARTMAN
Abstract. Let D(v) be the number of pairwise disjoint Steiner quadruple systems. A simple counting
argument shows that D(v) =< v 3. In this paper it is proved that D(2’n) >= (2
)n, k > 2, if there exists
a set of 3n pairwise disjoint Steiner quadruple systems of order 4n with a certain structure. This implies that
D(v) >_- v o(v) for infinitely many values of v. New lower bounds on D(v) for many values of v that are not
divisible by 4 are also given, and it is proved that D(v) >= 2 for all v 2 or 4(mod 6), v >= 8.
Key words. Steiner quadruple system, orthogonal array, one-factorization, disjoint Steiner systems
AMS(MOS) subject classifications. 51El0, 05B30, 05B15
1. Introduction. A Steiner quadruple system (SQS) is a pair (Q, q) where Q is a
finite set of points and q is a collection of 4-element subsets of Q called blocks such that
every 3-element subset of Q is contained in exactly one block of q. The number of points
in Q is the order ofthe SQS, and it is well known that an SQS of order v, denoted SQS (v),
has by
() blocks. Hanani [5] proved that Steiner quadruple systems of order v exist
if and only if v
2 or 4 (mod 6). Two SQS (Q, q and (Q, q2) are disjoint if q f3
A coloring of an SQS is a partition of the set of points into color classes such
q2
that no block is properly contained in any color class. An SQS is k-chromatic if it can
be k-colored, but no proper coloring having fewer than k color classes exists. A set of p
pairwise disjoint SQSs (PDQs) is mutually 2-chromatic if the same partition of Q is a 2coloring of all the PDQs. If(Q, q) is a 2-chromatic SQS(2v), with 2-coloring .4, B, then
by Doyen and Vandensavel [3], AI
[BI v, and the number of blocks nl, /72, /’/3
that meet A in 1, 2, and 3 points, respectively, is
.
n l=n3=
n2=
3
2"
Hence, the maximum size of a set of disjoint mutually 2-chromatic SQS(2v) is v since
the number of 4-subsets intersecting A in 3 points is ()v.
Let D(v) denote the maximum number of PDQs. Since () (v 3)by, we have
that D(v) _-< v 3. A set of v 3 PDQs of order v is called a large set. An application
of sets of PDQs is in the construction of constant weight codes with distance 4 ].
It is well known that D(4) 1, D(8) 2, and Kramer and Mesner [11] proved
that D(10) 5. Phelps 17 proved that D(2.5 ) -> 5 Phelps and Rosa 19 proved that
D(2.5 a. 13 b. 17 c) >_- 5 a. 13 17 c, for all a, b, c -> 0, a + b + c > 0, and Lindner [12]
proved that D(2v) >= v for v 2 or 4 (mod 6), v >_- 8. Recently, Phelps [18] has shown
that D(22) >_- 11. All the PDQs of these four constructions are mutually 2-chromatic.
Lindner [13] proved that D(4v) >- 3v for v
2 or 4 (mod 6), v >_- 8 by using his
v PDQs of order 2v 12 ].
In 2 we use a construction with a similar structure to the one of Lindner [13 to
obtain D(4v) >_- 3p, where v
or 5(mod 6) and a set ofp mutually 2-chromatic PDQs of
order 2v exists. If p v then our set of 3p PDQs is maximal.
In 3 we use the PDQs of Lindner [13 ], our PDQs of 2, and a set of 2
Boolean SQSs of order 2 to obtain D(2v) >_- (2
)v for v 2 or 4(mod 6), v >_- 8,
,
.
"
Received by the editors October 6, 1989; accepted for publication (in revised form) July 13, 1990.
Computer Science Department, Technion, Haifa, Israel.
IBM Israel, Science and Technology and Scientific Center, Haifa, Israel.
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